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We are concerned with the Keller--Segel--Navier--Stokes system begin{equation*} left{ begin{array}{ll} rho_t+ucdot ablarho=Deltarho- ablacdot(rho mathcal{S}(x,rho,c) abla c)-rho m, &!! (x,t)in Omegatimes (0,T), m_t+ucdot abla m=Delta m-rho m, &!! (x,t)in Omegatimes (0,T), c_t+ucdot abla c=Delta c-c+m, & !! (x,t)in Omegatimes (0,T), u_t+ (ucdot abla) u=Delta u- abla P+(rho+m) ablaphi,quad ablacdot u=0, &!! (x,t)in Omegatimes (0,T) end{array}right. end{equation*} subject to the boundary condition $( ablarho-rho mathcal{S}(x,rho,c) abla c)cdot u!!=! abla mcdot u= abla ccdot u=0, u=0$ in a bounded smooth domain $Omegasubsetmathbb R^3$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $mathcal{S}in C^2(overlineOmegatimes [0,infty)^2)^{3times 3}$ satisfies $|mathcal{S}(x,rho,c)|leq C_S $ for some $C_S>0$, and the initial data satisfy certain smallness conditions.
In this paper, we consider the following system $$left{begin{array}{ll} n_t+ucdot abla n&=Delta n- ablacdot(nmathcal{S}(| abla c|^2) abla c)-nm, c_t+ucdot abla c&=Delta c-c+m, m_t+ucdot abla m&=Delta m-mn, u_t&=Delta u+ abla P+(n+m) ablaPhi,qquad ab
In this paper, we proposed a coupled Patlak-Keller-Segel-Navier-Stokes system, which has dissipative free energy. On the plane $rr^2$, if the total mass of the cells is strictly less than $8pi$, classical solutions exist for any finite time, and thei
This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion begin{eqnarray} left{begin{array}{lll} n_t+ucdot abla n= ablacdot(| abla n|^{p-2} abla n)- ablacdot(nchi(c) abla c),& xinOmega, t>0, c_t+ucdot
We study the regularity and large-time behavior of a crowd of species driven by chemo-tactic interactions. What distinguishes the different species is the way they interact with the rest of the crowd: the collective motion is driven by different chem
In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Fu